(Lattice approximation) Does UV stability lead to continuum limit of a subsequence?

Question

In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like

Uniform boundedness of the partition function in the limit where lattice spacing goes to zero.

For example, Theorem 2 in this article deals with such an estimate.

However, I am confused if UV stability leads to existence of a continuum limit by compactness argument. The definition of continuum limit is from p.379 of this book. This definition pertains to Schwinger functions, so I guess the continuum functional measure should be somehow reconstructed by the moment problem.

Here are my main confusions:

First, the space of (tempered) distributions with the strong dual topology is a Montel space, implying that a bounded sequence has a convergent subsequence. So, some kind of uniform boundedness seems necessary.

Nevertheless, it is not clear to me if UV stability of lattice approximations implies boundedness in the relevant space of distributions. For example, does Theorem 2 in this article imply such estimate in the space of 3D periodic distributions?

It is stated in p.394-395 of this book that Balaban constructed continuum limit of 4D lattice pure YM theories. However, the papers cited there are devoted to UV stability without explicit mention of continuum limit, so I am confused.

I fear that this post is not detailed enough, but still hope for any answer from experts in this field.

Answer 1

I am not aware of a theorem which has UV stability as a hypothesis and delivers the construction of a continuum limit, even modulo taking subsequences. UV stability is just a combination of suitable upper and lower bounds on the partition function $Z$. If I remember correctly, this terminology and the emphasis on this notion is due to the Italian School of constructive QFT around Giovanni Gallavotti and Giuseppe Benfatto. See the article "Some probabilistic techniques in field theory" by Benfatto et al. where they show such a stability property for the hierarchical massive $\phi^4$ model in $2$ and $3$ dimensions. In "Ultraviolet stability in Euclidean scalar field theories" by the same authors, they do the same for the Euclidean rather than hierarchical model.

The importance of UV stability is that it is a milestone, and it should convince the few experts that an RG approach to constructing a continuum limit is feasible. But, there is still work to do after that. Both the above articles have a final section before the appendices called "Concluding remarks". If you read them you will see statements like: with a bit more effort one should be able to use our methods to prove bounds for $Z(f)$ which is the partition function in the presence of a source term given by a test function $f$. This last property, if proven, gets you closer to a continuum limit construction because $Z(if)/Z=:\Phi(f)$ is the characteristic function of the measure on Schwartz distributions one wants to take the limit of. Existence of subsequential limits is a consequence of tightness which is equivalent to the equicontinuity at the origin of this family of characteristic functions, indexed by the cut-offs.

In the book by Rivasseau, a similar milestone is emphasized: the construction of the pressure as a limit of $\log(Z)$ divided by the volume. This is a bit stronger than UV stability, or is a version of the latter where the gap between the upper and lower bounds on $Z$ vanishes in the limit say of large volume.

My personal opinion on the matter, is that going from an RG analysis for $Z$ to one for $Z(f)$ is a fundamental question, especially in the case where the source term is for a composite field like $\phi^2$ instead of $\phi$ itself. The main problem in constructive QFT from a more traditional RG-based point of view (as distinguished from say the new developments with the stochastic quantization approach) is to devise an RG analysis which can handle non-spatially uniform potentials, and frees itself from the requirement of translation invariance. The latter is why almost all the rigorous RG papers use periodic boundary conditions.

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Additional remarks following questions in the comments:

1. The probability measures here have expectations of the form $$ \mathbb{E}(\cdots)=\frac{1}{Z}\int \cdots e^{-V(\phi)} \ d\mu(\phi) $$ where $\mu$ is the Gaussian or free measure of reference (like the GFF) and $V$ is the potential, and $$ Z=\int e^{-V(\phi)} \ d\mu(\phi) $$ is the partition function to be bounded above and below in the UV stability statement. The potential is typically given by a sum of terms like $$ \lambda\int_{\mathbb{R}^d} \phi(x)^4\ d^dx $$ By non-spatially uniform potentials, I mean ones where we have more general terms like $$ \int_{\mathbb{R}^d} \lambda(x)\ \phi(x)^4\ d^dx\ . $$

2. The way I mentioned translation invariance (TI) has nothing to do with the goal of proving TI. I was talking about finding a renormalization group method which is applicable to non-spatially uniform potentials, and not limited in scope to the uniform ones only (where $\lambda(x)$ is constant with respect to the position $x$). If one uses lattice cutoffs (e.g., working on a box of linear size $2^N$ in the lattice $2^{-N}\mathbb{Z}^d$) and one's goal is to prove say the Osterwalder-Schrader Axioms, then to establish Euclidean invariance one needs to prove TI as well as rotation invariance (RI) after the limit $N\rightarrow\infty$ is taken. In this situation, proving TI is much less difficult than proving RI, especially if one takes periodic boundary conditions for the box.